A dynamic collective choice model with an advertiser

We consider within the framework of Mean Field Games theory a dynamic discrete choice model with an advertiser, where a large number of minor agents (e.g., consumers) are choosing between two predefined alternatives while influenced by social and advertisement effects. For example, in schools, teenagers' decisions to smoke are considerably affected by their peers (social effect), as well as the ministry of health campaigns against smoking (advertisement effect). The advertiser is “Stackelbergian”, in the sense that it makes its decision first and then the consumers make their choices. We show for a continuum of minor agents that there exists a Stackelberg solution. Moreover, when the minor agents are initially uniformly distributed on a line segment, we give an explicit form for the solution and characterize it by a scalar describing the way the population of agents splits between the destination points.